I know from two sources that it is (or at least was) unknown whether there are infinitely many geometrically distinct closed geodesics for every Riemannian metric on $S^3$, the 3-sphere (Weinberger, Computers, Rigidity, and Moduli, 1995, p.101; Berger, A Panoramic View of Riemannian Geometry, 2003, p.461). And from the same two sources, that it is known that there is at least one closed geodesic on any compact Riemannian manifold (Lusternick and Fet). My question is whether or not it is known that there are at least two distinct closed geodesics on $S^3$?
On $S^2$ it is known there are at least three simple closed geodesics (Lusternick and Schnirelmann), and infinitely many periodic geodesics (Bangert, Franks, Hingston). It might help an idea I'm considering if it were known there is more than one closed geodesic on $S^3$. Thanks for pointers!